extraDistr (version 1.9.1)

Pareto: Pareto distribution

Description

Density, distribution function, quantile function and random generation for the Pareto distribution.

Usage

dpareto(x, a = 1, b = 1, log = FALSE)
ppareto(q, a = 1, b = 1, lower.tail = TRUE, log.p = FALSE)
qpareto(p, a = 1, b = 1, lower.tail = TRUE, log.p = FALSE)
rpareto(n, a = 1, b = 1)

Arguments

x, q

vector of quantiles.

a, b

positive valued scale and location parameters.

log, log.p

logical; if TRUE, probabilities p are given as log(p).

lower.tail

logical; if TRUE (default), probabilities are \(P[X \le x]\) otherwise, \(P[X > x]\).

p

vector of probabilities.

n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability density function $$ f(x) = \frac{ab^a}{x^{a+1}} $$

Cumulative distribution function $$ F(x) = 1 - \left(\frac{b}{x}\right)^a $$

Quantile function $$ F^{-1}(p) = \frac{b}{(1-p)^{1-a}} $$

References

Krishnamoorthy, K. (2006). Handbook of Statistical Distributions with Applications. Chapman & Hall/CRC

Examples

x <- rpareto(1e5, 5, 16)
hist(x, 100, freq = FALSE)
curve(dpareto(x, 5, 16), 0, 200, col = "red", add = TRUE)
hist(ppareto(x, 5, 16))
plot(ecdf(x))
curve(ppareto(x, 5, 16), 0, 200, col = "red", lwd = 2, add = TRUE)